Patterns of humanity

I have mentioned a few times in previous posts about fractal images often being archetypal.  I’m not really sure I was correct in describing them that way, but what I will say is that they often look familiar in a way that transcends their immediate translation into a real object.  For instance, when I first looked at this fractal image, I immediately thought of the Aztecs and the patterns they incorporated into their art forms.  Having now looked at some Aztec art in more depth, I’m not even sure why I thought of them, other than the  feather motif and the colours.  That’s just what immediately came to mind.  And I have other fractal images created (not shown here) which really speak to me of Native American blanket patterns.  Others might look at this piece and be more immediately aware of the hourglass shape.  Maybe you will look at it and see something else entirely.

My point is, from near the beginning of humanity, we have been making patterns, whether or not they were drawn with a stick in the sand, sculpted along the edge of a stone building’s rooftop, painted on a cathedral ceiling, or digitally on a tablet.  Possibly for 60,000 years, we have been making patterns!  And if you look at all the patterns we have been making, you may notice that many of them are self-similar on smaller and smaller scales.  We were making fractals and we didn’t have a name for what was common to them all.  And we didn’t have a concept of the way fractals were involved in the geometry of nature – not consciously, anyway.  Maybe we were consciously inspired by nature, but didn’t recognize that specific aspect of it.  Only for the last 30-40 years have we, thanks to Mandelbrot, come to an awareness of this common denominator.  I like the way fractals connect all of humanity over time, and the way they connect us to nature.   I’m really looking forward to exploring this with future pieces!

Aztec Gold. Watercolour on Gessoed Paper. 20x20". $625.00. Lianne Todd

Aztec Gold.
Watercolour on Gessoed Paper.
20×20″.
$650.00.
Artist Lianne Todd

Advertisements

Physical Phenomena

If I am postulating that the universe is fractal in nature, it makes sense that structures formed by molecules behaving in their natural way should be recognizable as fractals.  Such is the case with frost, turbulence, and bubbles.  Just do a little Google search with each of those terms alongside fractal, and you’ll see what I mean.  Mandelbrot made groundbreaking progress modelling turbulence, which had confounded mathematicians before him, using fractal geometry.

It also makes sense that in my random wanderings through the fractal universes I create, I encounter images that remind me of these phenomena.  Such is the case with these two fractals which I chose to paint.  I especially like the way the bubbles in Turbulence & Bubbles look like they are in the process of being blown, sometimes from multiple locations, and melding together when they meet, just like real bubbles would.

Hot Frost. Watercolour on Aquabord. 6x6". $175.00 Lianne Todd

Hot Frost.
Watercolour on Aquabord.
6×6″.
$175.00
Lianne Todd

Turbulence & Bubbles. Watercolour on Gessoed Paper. 20x20". $625.00. Lianne Todd

Turbulence & Bubbles.
Watercolour on Gessoed Paper.
20×20″.
$650.00.
Lianne Todd

Imagining the Cosmos

As you may know, if you have been reading all of my blog posts, I like to dabble a little in cosmology.  Not that I really know anything about it, but it fascinates me, and I like the kind of abstract thought it stimulates.  I was given a Great Course one Christmas on Dark Matter, Dark Energy, The Dark Side of the Universe, which I’ve enjoyed a great deal.  Dr. Sean Carroll is great at explaining cosmology in a way that I, at least, can understand.  I also read Brian Greene’s The Hidden Reality, a fascinating book.  I subscribe to many Facebook Pages which post news about the latest pieces of knowledge in this field.  The abstract thought appeals to me, and I think I’m pretty good at it, but my math and physics skills are limited to helping out my children when they were doing grade twelve homework.  Which may be nothing to sneeze at, but is less than what is required for particle physics.

One of the things that really strikes me, is that the visual components, be they illustrations or actual data translated into an image, of almost every piece of news in cosmology, are recognizable to me as being fractal in nature.  Perhaps it is because I’ve spent so much time looking at fractals, zooming in, and examining them from every angle, that I notice this.  I am always mystified when no mention of fractals is made, in these cases.  I’ve written a  whole post (and another) about fractal dimensions before, so I won’t go into that here, but that’s another part of the puzzle I like to think about.

I know these two fractals don’t really illustrate anything in particular, but they make me think along the lines of particle physics, and stardust, and the early universe.

"Stardance". Watercolour on gessoed paper. 20x20". $625.00 Lianne Todd

“Stardance”. Watercolour on gessoed paper. 20×20″.
$650.00
Lianne Todd

"Particles and Fields". Fractal Digital Art on Metal, single edition print. 20x20". $325.00 Lianne Todd

“Particles and Fields”.
Fractal Digital Art on Metal, single edition print. 20×20″.
$345.00
Lianne Todd

This past summer, during my exhibit, I was told about a person in the same city (London, Ontario) who, it seemed, had a lot of the same thoughts I was having about the fractal nature of the universe.  She is a software engineer and has been studying fractals for much longer than I have!  Needless to say, her math skills are much better than mine.  I was fortunate to meet her (her name is Lori Gardi) this fall.  She has two websites, the first of which I’m going to direct you to Here, in which she has laid out some of her thoughts in a pretty clear way.  I’ll link you to the more recent one later… I think it’s a good idea to start at the beginning of her thought process. We (artists, software engineers, mathematicians, physicists, philosophers) may not all have the same thought processes or reach the same conclusions in our explorations of fractals and their role in the universe, but all avenues should be explored as long as they can exist within the rules that have been truly established by scientists and mathematicians in the past.  I am, unfortunately, not capable of judging whether anything follows those rules, but others who can, need to at least look at this work with an open mind and decide for themselves. ESPECIALLY if it may help solve any of the mysteries still out there.

Further into the realm of Fantasy

Often, when I’m voyaging through the through the little fractal universes I have generated using the software which I am so thankful exists, I encounter ‘places’ that look like they belong in an illustration for a book I’ve read somewhere along the way. I also encounter characters that look like they belong in those places.  Such was the case for this piece:

The Mage Emerges Digital Art Printed on Metal, single edition 24x24" $425.00 Lianne Todd

The Mage Emerges
Digital Art Printed on Metal, single edition
24×24″
SOLD.  Private collection.
Artist Lianne Todd

The Landing

Congratulations to the European Space Agency, and to all of humanity, for boldly going where no one has gone before… for landing the human-built robot Philae on a comet!!
This is truly a historic moment and an example of how our imaginations can lead us to realities our ancestors may have considered impossibilities.  It will lead to a greater understanding of our solar system and of the universe.

In honour of this event, I’m posting my piece called The Landing.  I can’t think of a better day to do it!  It’s purely imaginary, constructed of three different fractals, and belongs in the realm of fantasy or science fiction.  But then, not so long ago, so did landing on a comet…

The Landing Digital Art printed on metal, single edition 20x20" Lianne Todd $325.00

The Landing
Digital Art printed on metal, single edition
20×20″
Artist Lianne Todd
SOLD.  Private Collection.

 

The Experiment

This series of four paintings was an experiment.  Not a very scientific one but I did try to control variables and make predictions.  My hypothesis was that since all pigments are different in their molecule size, shape, and hydrophilic and hydrophobic qualities, they would all move differently through the medium of water, and as they interacted with each other, and that their movement would be fractal.  In other words, I expected them to appear, at the end, as if they were something like a cloud, or some other natural item that is already known to be able to be modelled using fractal geometry.  A coastline, for instance.

All four pieces were executed and controlled in the same way (I won’t give away all my secrets!), the only differences being the pigments I used and the order they were used in.  I tried to reduce the effects of gravity by levelling my table but it is kind of obvious there was a tiny bit of gravitational effect.  I changed the orientation of the final products so that when they were hung together, it would be aesthetically pleasing.  Other than that, the results you see here are basically the raw data.

I called them Negative Nebulae, because I looked at all the white space around them and imagined if it was black, and the colours were reversed, they would look a bit like those photos you see from NASA of distant nebulae.  In other words, these would be like the negatives of those photos.

Here they are – what do you think?:

Nebula Negative I Watercolour on Yupo 10x10" Lianne Todd

Nebula Negative I
Watercolour on Yupo
10×10″ (sold)
Lianne Todd

Nebula Negative II Watercolour on Yupo 10x10" Lianne Todd

Nebula Negative II
Watercolour on Yupo
10×10″ $125.00
Lianne Todd

Negative Nebula III Watercolour on Yupo 10x10" Lianne Todd

Negative Nebula III
Watercolour on Yupo
10×10″ (sold)
Lianne Todd

Negative Nebula IV Watercolour on Yupo 10x10" Lianne Todd

Negative Nebula IV
Watercolour on Yupo
10×10″ (sold)
Lianne Todd

Here is what they looked like at the exhibit:

KONICA MINOLTA DIGITAL CAMERA

 

 

Curious to know what they do really look like when you invert the colours?

Non Negative Nebula I (the inversion) Lianne Todd

Non Negative Nebula I
(the inversion)
Lianne Todd

Non Negative Nebula II (the Inversion) Lianne Todd

Non Negative Nebula II
(the Inversion)
Lianne Todd

Non Negative Nebula III (the inversion) Lianne Todd

Non Negative Nebula III
(the inversion)
Lianne Todd

Non Negative Nebula IV (the inversion) Lianne Todd

Non Negative Nebula IV
(the inversion)
Lianne Todd

 

Fried Eggs

This is just one of my favourites.  It’s only little, 6×6″, but like most fractals it took a long time to paint.  So many tiny little fried eggs!  It’s framed in a black lacquered shadow box frame so that it floats in the frame.  The total size, frame and all, is roughly 12×12″.

Yet again, we see a natural shape.  Well, natural, in that we are natural and we naturally like to fry eggs. Sometimes I find that it isn’t so much the repetition on smaller and smaller scales that makes me think of natural objects or phenomena when I look at fractals, but the shape that is being repeated.

As usual, it’s copyrighted and watermarked.

Fried Eggs Lianne Todd Watercolour on Aquabord 6x6" $175.00

Fried Eggs
Collection the Artist Lianne Todd
Watercolour on Aquabord
6×6″

Patterns

This piece is called Looking Through.

"Looking Through" Digital Art printed on metal, single edition 20x20" $325.00 Artist: Lianne Todd

“Looking Through”
Digital Art printed on metal, single edition
20×20″
$345.00 Artist: Lianne Todd

Looking through what?  A microscope?  A telescope? A porthole?

In a fractal universe, it doesn’t really matter.  Similar patterns are present on multiple scales.  Use your imagination!

This image is actually a combination of fractals – one for the thing we are looking through (the self-similarity on smaller scales provides the illusion of perspective and depth here), and one for what we are looking at (this is a flame fractal – more about them later).

As illustrated here, fractal geometry is quite versatile.  I’ve seen some discussion on ‘true’ fractals versus ‘near’ fractals and I would like to address that here for a moment.  There seems to be an opinion out there that for a fractal to be ‘true’ it must be a)infinite and b)exactly the same no matter what scale you look at.  Having read most of Benoit Mandelbrot’s Fractal Geometry of Nature, I have a problem with these stipulations.  First of all, the equation for Mandelbrot’s set is z_{n+1}=z_n^2+c, with as the number of iterations, where c is a complex parameter.  

There is more to explaining the Mandelbrot set than that, of course, but that is the equation, and if n is a given number, then it’s not infinite, is it?  Perhaps the possibility of an infinite number of iterations exists, but that’s an argument for another day.

And even Mandelbrot’s set is not exactly the same on multiple scales. The PATTERN is there, it’s just slightly altered at different scales.  It is self-similar.  This is one of the things which makes fractal geometry so suitable for modelling the universe.

In my understanding, there was never a suggestion by Mandelbrot, the founder of fractal geometry, that a “true” fractal had to be infinite OR exactly the same on multiple scales.  Rather, a fractal is strictly defined as “a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.”

So, perhaps I’m getting it all wrong, but if you would like to argue I would welcome your discussion.

And now, because I was once a biologist and if you’re anything like me you need a more highly magnified look at that thing, here is a zoom of what you were “looking through” at:

sea creature zoomed in

 

The Photographs

It isn’t difficult to spot natural fractals all around you, if you know what you’re looking for.  It’s quite probable that you just don’t recognize them because you haven’t looked at enough computer generated fractals, at enough scales, to realize that even if something in nature doesn’t look like a whole fractal, i.e., you don’t really see the repetition of a pattern on smaller and smaller scales,  it will look like part of one.

Before Benoit Mandelbrot came along, nature was regarded as a rather chaotically influenced version of Euclidian geometry.  The artist Paul Cezanne said as instruction to young painters: “Everything in Nature can be viewed in terms of cones, cylinders, and spheres.”  But Mandelbrot’s famous quote “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line” contradicts this, and rings very true. 

Some mathematical experts are able to generate entire landscapes, or, small parts of nature like a fern leaf, just using fractal formulas.  Benoit Mandelbrot gave a few examples of these in his book, The Fractal Geometry of Nature.  Indeed, the generation of natural looking landscapes, textures, etc. using fractals is quite common, in video games.  Ever wondered how the game manages to keep up the appearance of the surrounding landscape the character is travelling through?  That’s how.  They are also used in movies, creating alien landscapes.

Some examples of fractals in nature are depicted in the photographs that were part of my exhibit this summer.  The patterns of ice crystal formation, mountain ranges, clouds, branching patterns of trees, growth patterns of mosses and lichens, flower structure, butterfly wings and their coloration, fur growth and patterning… on every scale you can recognize fractals, not just on earth but in the entire solar system and universe.  (More about that later).  For now, have a look at these, and look at the world in a new way the next time you go outside.

Top to Bottom, Left to Right: Freezing; Mountains & Vapour; Mossy Branches; Reiterated Beauty; Nature's Drapery; All the Markings of a Bandit.  Digital Photography.  10x10" Prints, framed  $125.00

Top to Bottom, Left to Right:
Freezing; Mountains & Vapour; Mossy Branches; Reiterated Beauty; Nature’s Drapery; All the Markings of a Bandit. Digital Photography. 10×10″ Prints, framed $125.00

As they appeared in the exhibit.

As they appeared in the exhibit at The ARTS Project.  Thanks again to the Ontario Arts Council!

Butterflies and Moths

Insects, and particularly butterflies and moths, are recurring motifs that I often encounter when I’m creating fractals.  Sometimes, it’s just the simple shape, and other times it seems to be a whole detailed creature.  Sometimes it’s done with what I call the ‘regular’ fractal generator and other times with the flame fractal generator (more on those differences later).  If a mathematical formula iterated over and over by a computer can randomly generate images like these in a matter of minutes or hours, imagine what the physical forces of nature and a few billion years of evolution can do with a periodic table of elements (and, shall I say, an underlying fractal structure?).  Oh wait, you don’t have to imagine.  You can go outside!

 

(All images are watermarked and copyrighted)

Butterfly Hub Digital Art printed on metal, single edition 20x20" $325.00

Butterfly Hub – Artist Lianne Todd
Digital Art printed on metal, single edition
20×20″
$345.00

Detail of Butterfly Hub

Detail of Butterfly Hub

Butterflire - Artist Lianne Todd Digital Art printed on metal, single edition 20x20" $325.00

Butterflire – Artist Lianne Todd
Digital Art printed on metal, single edition
20×20″
$345.00

Detail of Butterflire

Detail of Butterflire

Mother of Moths - Artist Lianne Todd Digital Art printed on metal, single edition 12x12" SOLD

Mother of Moths – Artist Lianne Todd
Digital Art printed on metal, single edition
12×12″
SOLD. Private Collection.

Pollinator - Artist Lianne Todd Digital Art printed on metal, single edition 16x16" $225.00

Pollinator – Artist Lianne Todd
Digital Art printed on metal, single edition
16×16″
$240.00

A thumbnail of the raw generated fractal - just to illustrate part of the process.

A thumbnail of the raw generated fractal – just to illustrate part of the process.